Time Decay
Now, in the present situation the wave function decays in time rather than space, but
the argument is very similar. To construct the decaying wave function we must add
together plane waves in time
corresponding to different energies. The
requir
Einsteins Solution of the Specific Heat Puzzle
The simple harmonic oscillator, a nonrelativistic particle in a potential
is an
excellent model for a wide range of systems in nature. In fact, not long after Plancks
discovery that the black body radiation s
Cauchys theorem has a very important consequence: for an integral from,
say, za to zb in the complex plane, moving the contour in a region where the function is
analytic will not affect the result, because the difference between the integral over the
orig
Saddlepoint Estimation of n!
We use the identity
To picture tne-t, here it is for n = 10:
Note that
Therefore, in the neighborhood of the maximum value of f(t) at t = n,
For integer n, the function is analytic in any finite region of the complex plane.
Ta
Rapidly Oscillating Integrals
How to evaluate
in an unambiguous fashion: an introduction to moving the
contour of integration and the Method of Steepest Descent.
The familiar Gaussian integral
is easy to understand. Plotting the
integrand, (here for a = 1
Operator Approach
Having scaled the position coordinate x to the dimensionless
let us also scale the momentum from p to
(so
).
The Hamiltonian is
Dirac had the brilliant idea of factorizing this expression: the obvious
thought
isnt quite right, because it
Higher Energy States
It is clear from the above discussion of the ground state that
is the natural
unit of length in this problem, and
that of energy, so to investigate higher energy
states we reformulate in dimensionless variables,
.
Schrdingers equation
Schrdingers Equation and the Ground State Wave Function
From the classical expression for total energy given above, the Schrdinger equation
for the quantum oscillator follows in standard fashion:
What will the solutions to this Schrdinger equation look li
Model of a Decaying State
The momentum-position uncertainty principle
has an energy-time
analog,
. Evidently, though, this must be a different kind of relationship to
the momentum-position one, because t is not a dynamical variable, so this cant have
anyt
creation operator
It is easy to check that the state
is an eigenstate with eigenvalue
provided it
is nonzero, so the operator a takes us down the ladder. However, this cannot go on
indefinitelywe have established that N cannot have negative eigenvalues. W
Contour Integration: Cauchys Theorem
Cauchys theorem states that the integral of a function of a complex variable around a
closed contour in the complex plane is zero if the function is analytic in the region
enclosed by the contour.
This theorem can be p
Analytic Functions
Suppose we have a complex function f = u + iv of a complex variable z = x + iy,
defined in some region of the complex plane, where u, v, x, y are real. That is to say,
with u(x,y) and v(x,y) real functions in the plane.
We now assume th